Theorem bj-sb56 32639

Description: Proof of sb56 2150 from Tarski, ax-10 2019 (modal5) and bj-ax12 32634. (Contributed by BJ, 29-Dec-2020.) (Proof modification is discouraged.)

Assertion

Ref	Expression
bj-sb56	|- ( E. x ( x = y /\ ph ) <-> A. x ( x = y -> ph ) )

Distinct variable group:   x, y

Allowed substitution hints:   ph( x, y)

Proof of Theorem bj-sb56

Step	Hyp	Ref	Expression
1		bj-ax12 32634	. . . 4 |- A. x ( x = y -> ( ph -> A. x ( x = y -> ph ) ) )
2		pm3.31 461	. . . . 5 |- ( ( x = y -> ( ph -> A. x ( x = y -> ph ) ) ) -> ( ( x = y /\ ph ) -> A. x ( x = y -> ph ) ) )
3	2	aleximi 1759	. . . 4 |- ( A. x ( x = y -> ( ph -> A. x ( x = y -> ph ) ) ) -> ( E. x ( x = y /\ ph ) -> E. x A. x ( x = y -> ph ) ) )
4	1, 3	ax-mp 5	. . 3 |- ( E. x ( x = y /\ ph ) -> E. x A. x ( x = y -> ph ) )
5		bj-modal5e 32636	. . 3 |- ( E. x A. x ( x = y -> ph ) -> A. x ( x = y -> ph ) )
6	4, 5	syl 17	. 2 |- ( E. x ( x = y /\ ph ) -> A. x ( x = y -> ph ) )
7		equs4v 1930	. 2 |- ( A. x ( x = y -> ph ) -> E. x ( x = y /\ ph ) )
8	6, 7	impbii 199	1 |- ( E. x ( x = y /\ ph ) <-> A. x ( x = y -> ph ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   A.wal 1481   E.wex 1704

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-10 2019  ax-12 2047

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1705

This theorem is referenced by:  bj-dfssb2  32640